tween 2 mm and 8 mm determined by sieve analy-
p3
λ2
=
ses on the samples. Beltaos and Dean (1981) re-
(27)
1 - p Kgds
ported smaller diameter particles (1 mm to 6 mm)
where λ is the seepage coefficient, K is a dimen-
in a frazil deposit on the Smoky River, Alberta,
sionless seepage coefficient, and ds is a variable
with 60% by weight in the range of 1.1 to 2.4 mm.
describing the particles (ds = 6/Ms, where Ms is
Shen and Wang (1992) reported mean particle di-
the specific area per unit volume of the solid par-
ameter of about 1 cm on the Yellow River, China.
ticles, or the ratio of surface area to volume). In
The fall velocity coefficient (Cf) for particles 5 mm
tests with square polyethylene blocks, Wong et al.
to 1 cm in diameter has been reported to be about
(1985) found that K varied from 0.3 to 0.7, with
1.0 (Shen and Wang 1995).
the higher value for randomly placed blocks.
Beltaos (1999) suggested that ds can be approxi-
mated as twice the average ice thickness of the ice
MISCELLANEOUS VARIABLES AFFECTING
blocks. This approach was used by Lever and
RIVER ICE HYDRAULICS
Gooch (1998), who obtained careful water level
measurements and discharge estimates at rated
Anchor ice growth
cross sections for a breakup jam thought to be
Anchor ice growth occurs when frazil ice ad-
grounded at the toe. Assuming K = 0.6 and ds ≈
heres to rocks or other materials on the bed of a
twice the average parent ice thickness, they calcu-
river. The presence of anchor ice, or frazil ice ac-
lated p = 0.70 0.2. However, this high value for
cumulated on submerged surfaces, can affect river
breakup jam porosity may be a result of incom-
hydraulics in a number of ways, through increas-
plete grounding or piping through the jam, be-
ing the effective bed level and changing the effec-
cause the analysis includes only porous flow.
tive bed roughness, among others. Brown et al.
(1953) reported that frazil ice adhered first on the
Shear stress on the underside of the ice cover (τ)
upstream face of stones on the riverbed, or at the
As noted in eq 18, the shear stress on the un-
point of attachment to the bed for other submerged
derside of the ice cover, τ, is a function of ρ, g, Sf,
objects. Measurements showed that the average
and the variables necessary to calculate Ri: q, ni,
velocity at the attachment points on the upstream
face of the stones was 0.40 m/s compared to 0.77
nb, and nc. Based on observed water profiles,
Andres (1980) calculated τ = 18.6 Pa for the 1978
m/s along the sides (where anchor ice accumu-
lated later). Under favorable conditions the anchor
Athabasca River (Alberta) breakup jam. Andres
ice grew both horizontally and vertically to form
and Doyle (1984) report a range of 9.6 to 28.5 Pa
a mat of anchor ice. Anchor ice thickness appeared
and an average of 17.9 Pa for shear forces experi-
to be a function of water depth, temperature, and
enced during breakup jams on the Athabasca River
velocity, and reached 0.6 m in one location.
(Alberta) in 1977, 1978, and 1979. Rivard et al.
(1984) calculated τ = 7.21 Pa and τ = 10.0 Pa for
Data from Hirayama et al. (1997) from the
two cross sections of the 1983 Mackenzie River
Niuppu River, Japan, indicate an apparent aver-
age density of about 0.5 g/cm3 for anchor ice. They
(Northwest Territories) ice jam. Dean (1986) esti-
mated shear beneath frazil ice deposits to be about
also reported that the density of anchor ice appears
8 to 10 Pa during steady midwinter flow and 60 to
to increase slightly as water velocity increases.
Using a six-hour freezing index (FI, units C hr)
700 Pa during a severe water level change.
for the hours 0000 to 0600, they measured an av-
erage anchor ice accumulation rate of 0.10 m3 per
Frazil particle diameter (d) and fall
C hr. Anchor ice accumulation was observed to
velocity coefficient (Cf)
Few values of frazil particle sizes exist outside
be related to Froude number, F, as well as FI. Ac-
Alaska. Gosink and Osterkamp (1983) reported
cumulations occurred for 0.2 < F < 1.5, but none
frazil particle diameters ranging between 1 and 6
were seen for F < 0.2. A high FI (i.e., more intense
mm during a series of tests on the Chatanika River,
cold) allowed frazil ice to accumulate at lower
Alaska. Chacho et al. (1986) reported frazil diam-
values of F, but lower FI required a higher F be-
eters of 5 mm to 150 mm in the Tanana River. White
fore anchor ice accumulation was seen. Finally,
and Lawson (1992) cited unpublished data by
they noted that more anchor ice volume results on
Lawson and Brockett that report particles between
surfaces with higher coefficients of static friction.
1 mm and 25 mm in diameter, also on the Tanana
Kerr et al. (1997, 1998) investigated anchor ice
River, Alaska. They found mean particle sizes be-
growth on rounded gravel averaging 4.45 and
19
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